Finite Group Databasebeta

← #13147415

Subgroups — #13147415

Order 1728.

559 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88748C211yesmaximal, characteristic
2#11722448C211yesmaximal, characteristic, complemented
3#11722445C211yesmaximal, characteristic, complemented
4#11720791C214yesmaximal, complemented
5C2×Q8×H(3,𝔽3)C411yescharacteristic, complemented
6#88748—21no—
7#88748—21no—
8#88748C412yescomplemented
9#88748C2212yes—
10C2×C4⋊#272—21no—
11C2×C4⋊#272C411yescharacteristic, complemented
12C2×C4×#272—21no—
13C2×C4×#272C2211yescharacteristic
14C22⋊#1088—21no—
15#88631—21no—
16C4⋊#1088—21no—
17C4×#1088C2212yescomplemented
18#88627C2212yescomplemented
19#88626—21no—
20C2×C3×#239—62no—
21C22×#272—21no—
22C22×#272D411yescharacteristic
23C2×#1088—22no—
24Q8×H(3,𝔽3)—21no—
25Q8×H(3,𝔽3)C2×C412yes—
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)C2×C411yescharacteristic
28C2×#1084D411yescharacteristic
29C4⋊#272—22no—
30C4⋊#272—22no—
31C4⋊#272—21no—
32C4⋊#272C2×C412yes—
33C4×#272—22no—
34C4×#272—22no—
35C4×#272C2×C411yescharacteristic
36C4×#272D411yescharacteristic
37C4×#272D411yescharacteristic
38C4×#272C2311yescharacteristic
39#6188—22no—
40C3×#421—91nomaximal
41C2×C32×Q8—22no—
42C3×#239—64no—
43C3×#239—64no—
44C3×#239—64no—
45C3×#239—64no—
46C2×C3×D12—62no—
47C2×C3×D12—64no—
48C2×C3×C4×D3—62no—
49C2×C3×C4×D3—64no—
50C22×H(3,𝔽3)C22⋊C411yescharacteristic
51C2×#272—22no—
52C2×#272—41no—
53C2×#272—42no—
54C2×#272C22⋊C411yescharacteristic
55C2×#272C2×D411yescharacteristic
56#1088—42no—
57C4×H(3,𝔽3)—22no—
58C4×H(3,𝔽3)C22⋊C411yescharacteristic
59C4×H(3,𝔽3)C22×C411yescharacteristic
60#1084C22⋊C411yescharacteristic
61#1084C2×D411yescharacteristic
62C2×C3×#41—91no—
63C2×#239—62no—
64C2×C3×M16—91no—
65C3×#117—91no—
66C3×#104—94no—
67C22×C3×D3—62no—
68C22×C3×D3—64no—
69C32×Q8—24no—
70C32×Q8—24no—
71C2×C32×C4—24no—
72C2×C32×C4—22no—
73C2×C3×Dic3—62no—
74C3×D12—64no—
75C3×D12—68no—
76C3×D12—68no—
77C3×D12—64no—
78C3×C4×D3—62no—
79C3×C4×D3—62no—
80C3×C4×D3—62no—
81C3×C4×D3—62no—
82C3×C4×D3—68no—
83C3×C4×D3—68no—
84#421—91noSylow
85C2×H(3,𝔽3)—21no—
86C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
87#272—21no—
88#272—21no—
89#272—42no—
90C3×#41—92no—
91C3×#41—92no—
92C3×#41—181no—
93C3×#41—181no—
94C2×C3×Q8—62no—
95C2×C3×Q8#15711yescharacteristic
96C2×C3×D4—91no—
97C2×C3×D4—181no—
98C22×C3×C4—91no—
99C22×C3×C4—181no—
100#239—64no—
101#239—64no—
102#239—64no—
103#239—64no—
104C2×D12—62no—
105C2×D12—64no—
106C2×C4×D3—62no—
107C2×C4×D3—64no—
108C3×M16—92no—
109C3×M16—181no—
110C2×C3×C8—181no—
111C3×C4⋊C4—181no—
112C3×C22⋊C4—181no—
113C3×C42—92no—
114C22×C32—22no—
115C2×C3×D3—62no—
116C2×C3×D3—62no—
117C2×C3×D3—68no—
118C2×C3×D3—124no—
119C2×C3×D3—128no—
120C32×C4—28no—
121C32×C4—22no—
122C32×C4—22no—
123C3×Dic3—62no—
124C3×Dic3—62no—
125C2×#41—91no—
126C2×M16—91no—
127#117—91no—
128#104—94no—
129H(3,𝔽3)#42111yescharacteristic, complemented, Sylow
130C23×C3—91no—
131C23×C3—181no—
132C22×D3—62no—
133C22×D3—64no—
134C3×Q8—21no—
135C3×Q8—64no—
136C3×Q8—64no—
137C3×Q8C2×#15712yes—
138C3×D4—92no—
139C3×D4—182no—
140C3×D4—182no—
141C3×D4—181no—
142C2×C3×C4—21no—
143C2×C3×C4—64no—
144C2×C3×C4—62no—
145C2×C3×C4—91no—
146C2×C3×C4—91no—
147C2×C3×C4—91no—
148C2×C3×C4—91no—
149C2×C3×C4—91no—
150C2×C3×C4—182no—
151C2×C3×C4—182no—
152C2×C3×C4—182no—
153C2×C3×C4C2×#15711yescharacteristic
154C2×Dic3—62no—
155D12—64no—
156D12—68no—
157D12—68no—
158D12—64no—
159C4×D3—62no—
160C4×D3—62no—
161C4×D3—62no—
162C4×D3—62no—
163C4×D3—68no—
164C4×D3—68no—
165C3×C8—182no—
166C2×C32—24no—
167C2×C32—22no—
168C3×D3—122no—
169C3×D3—122no—
170C3×D3—128no—
171#41—92no—
172#41—92no—
173#41—181no—
174#41—181no—
175C2×Q8#108811yescharacteristic, complemented
176C2×D4—91no—
177C2×D4—181no—
178C22×C4—91no—
179C22×C4—181no—
180M16—92no—
181M16—181no—
182C2×C8—181no—
183C4⋊C4—181no—
184C22⋊C4—181no—
185C42—92no—
186C22×C3—62no—
187C22×C3—91no—
188C22×C3—91no—
189C22×C3—182no—
190C22×C3—361no—
191C22×C3—362no—
192C22×C3C22⋊#15711yescharacteristic
193C2×D3—62no—
194C2×D3—62no—
195C2×D3—68no—
196C2×D3—124no—
197C2×D3—128no—
198C3×C4—22no—
199C3×C4—68no—
200C3×C4—62no—
201C3×C4—62no—
202C3×C4—91no—
203C3×C4—91no—
204C3×C4—362no—
205C3×C4C22⋊#15711yescharacteristic
206C3×C4C22×#15711yescharacteristic
207Dic3—62no—
208Dic3—62no—
209C32—22no—
210C23—91no—
211C23—181no—
212Q8—21no—
213Q8C2×#108812yes—
214D4—92no—
215D4—182no—
216D4—182no—
217D4—181no—
218C2×C4—21no—
219C2×C4—91no—
220C2×C4—91no—
221C2×C4—91no—
222C2×C4—91no—
223C2×C4—91no—
224C2×C4—182no—
225C2×C4—182no—
226C2×C4—182no—
227C2×C4C2×#108811yescharacteristic
228C8—182no—
229C2×C3—21no—
230C2×C3—64no—
231C2×C3—62no—
232C2×C3—181no—
233C2×C3—181no—
234C2×C3—362no—
235C2×C3C2×C22⋊#15711yescharacteristic
236D3—122no—
237D3—122no—
238D3—128no—
239C22—91no—
240C22—91no—
241C22—182no—
242C22—361no—
243C22—362no—
244C22C22⋊#108811yescharacteristic
245C4—22no—
246C4—91no—
247C4—91no—
248C4—362no—
249C4C22⋊#108811yescharacteristic
250C4C22×#108811yescharacteristic
251C3—62no—
252C3#1059383111yescharacteristic
253C2—21no—
254C2—181no—
255C2—181no—
256C2—362no—
257C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, socleC2×C3C2×C22⋊#157
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC3×C4C22×#157
FittingC2×Q8×H(3,𝔽3)C4
prime radicalC2×#88748C2
cosocleC4×#272C23
hypercenterC2×C3×Q8#157
hypocenterH(3,𝔽3)#421
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11722448C2semidirect
3. #11722445C2semidirect
4. #11720791C2semidirect
5. C2×Q8×H(3,𝔽3)C4semidirect
8. #88748C4semidirect
11. C2×C4⋊#272C4semidirect
17. C4×#1088C22semidirect
18. #88627C22semidirect
129. H(3,𝔽3)#421semidirect
175. C2×Q8#1088semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#887482
#117224482
#117224452
#117207912
C3×#4219